Trường Đại học Bách khoa Tp COMPARING TWO POWER FLOW OPTIMIZATION ALGORITHMS BY ESTIMATING TECHNICO ECONOMIC EQUIVALENT STATES Luu Huu Vinh Quang Power System Department University of Technology – Vie[.]
COMPARING TWO POWER FLOW OPTIMIZATION ALGORITHMS BY ESTIMATING TECHNICO-ECONOMIC EQUIVALENT STATES Luu Huu Vinh Quang Power System Department University of Technology – VietNam National University – HCM City ABSTRACT This paper mentions the modified bus impedence matrix model applying to power flow optimization, and compares this model with the augmented rectangular system model with an eigenimage matrix The modified expressions to compute the bus PV voltages are shown and a new p_power loss coefficient expression is proposed for the power flow optimization computation To compare the different models of power flow optimization we propose a new modified estimative matrix for a method estimating the technico-economic equivalent states, and appreciating the optimum conditions of p_power generation between electric generators in a power system using the largest eigenvalue of an estimative matrix relating to the p_power loss coefficients and the system economic incremental fuel cost INTRODUCTION Power flow is the most frequently performed study for power system The power flow problem involves determining voltage and line flow, in a large electrical network, for a given load and generation schedule Many important contributions have been made on this along the years Any improvement to its method on storage, convergence, and reliability is of value The bus impedence matrix (zbus) power flow model is one of the known models for power flow calculation In this paper we present a modified zbus power flow algorithm and its development applying in power flow optimization The combination of many agents will cause not only one optimum operation point but also a technico-economic equivalent area where exist many technico-economic equivalent operation points These agents, such as generator fuel cost functions, the bus load powers, the network resistances, inductances and capacitances, etc are approximately determined, and the optimization algorithm with some simplified assumtion can result in small errors We will compare new loss coefficient expressions that are developed in the two p_power optimization algorithms, in which the first is related to modified zbus power flow model, the second is related to eigen-image vector power flow model [1] The method introduced in [2] is used to determine the size of the technico-economic equivalent area, therefore, we can compare the two methods by estimating the accuracy of the realization of optimum condition in power system operation MODIFIED ZBUS MODEL P_POWER FLOW OPTIMIZATION AND Let’s examine an electrical power system comprising m generators In all practical cases, the fuel cost of generator i can be represented as quadratic function Ci of p_power generation Pig Ci= aiP2ig + biPig + ci ; (1) i=1,2,…,m The economic p_power dispatching problem is to minimize the overall generating cost Ci In case of applying the zbus model, it can develop into the following equations [3] (2) a1 B11 B21 B12 a2 B22 B1m B2m : : : : : : : : Bm1 Bm2 am Bmm P1g P2g : : : Pmg b1 b2 bm Bo1 Bo2 : : : ; Bom where Pig is generator p_power; is system incremental fuel cost; Bij are the loss coefficients [3] (3) Bij (4) Nb Rk k 1(0.5(Vi Boi N j Vj )) (Re[C ki ] Re[C kj ] Im[C ki ] Im[C kj ]) ; BijPLj ; (5) N j Vo ( Pj TR( k Vi( k ) N Vo ( Pj TX( k ) j 1) TX( k 1) ) 1) Q (jk ) TR( k ) ) Vj( k ; 1) (6) (k) i j arctan ( Pj TX( k N Vo j 1) TR(k 1) rij cos (8) TX(k rij sin 1) Q (jk ( Pj TR( k (7) 1) 1) TR( k Q (jk ( k 1) j ( k 1) j 1) 1) TX( k ) 1) ; ) ( k 1) ; j ( k 1) ; j x ij sin x ij cos where k is the number of iteration; Vi, i are the bus voltage modul and argument; Vo is the modul of slack bus voltage; N is the independent bus quantity of electric network; rij, xij are the real and imaginary elements of zbus matrix; Pj, Q(k-1)j are the active and reactive bus powers; During iterative calculation, the bus q_power values are specified at the PQ bus, (Q(k-1)j=Q(o)j), and the new modified expressions to adjust the reactive power value at the PV bus j are proposed as follow (9) Q(jk ) (10) Ip (11) Pj sin Iq Vj(k 1) ( k 1) Ip j Vie sin Pj cos ( k 1) j ( k 1) j Q (jk Vj(k Vje The load flow calculation is to determine voltage and line flow, in a large electrical network, for a given load and generation schedule Using the zbus matrix model [4],[5],[6],[8] we can obtain the expressions to compute the bus voltage as follow Q (jk Vj( k N where Nb is the branch quantity of network; N is the bus number quantity of network; Rk is the resistance of branch k linking two buses i and j; Vi, Vj are the voltages at buses i, j ; Re[Cki], Re[Ckj], Im[Cki], Im[Ckj] are the real and imaginery parts of elements of current distribution factor matrix C; PLj is the p_power load at bus j We can carry out the power system steadystate optimization as shown in figure 1) 1) Q (jk Vj(k cos ( k 1) ; Iq j Vie cos 1) sin 1) ( k 1) j ; ( k 1) j 1) rjjVj(k 1) sin ( k 1) j x jjVj(k 1) cos z 2jj ( k 1) j ; where Vje is the expected voltage modul at the bus j; The bus q_power at the PV bus j must be satisfied (12) Qj Q(jk 1) Q(jk 1) Qj ; if the inequality (12) is not satisfied for the bus j, then the bus reactive power must be fixed at the limit, and this bus can be traited as a PQ bus in the next iteration In the case, to control the voltage at PV bus j, the expected bus voltage at this bus can be adjusted so that an inequality is satisfied (13) Vj Vj(k 1) ( k 1) j (Vj(k 1) )2 rjj cos Re[J iT ] (t m 1)(Vj cos jG m (t 2m 1)(Vi cos i G m Q (jk 1) Vj ; Re[J jT ] (17) Im[J jT ] Vj sin jG m ) Vi sin i G m ) ; (1 t m )(Vi cos i G m Vi sin i Bm ) ; (1 t m )(Vi cos i Bm Vi sin i G m ) ; where Re[JiT], Im[JiT], Re[JjT], Im[JjT] are the real and imaginary parts of injected transfomodel currents at the buses i and j; tm is the perunit transfo-ratio ; Gm, Bm are the real and imaginary parts of admittance of transformer branch m; AUGMENTED RECTANGULAR SYSTEM MODEL WITH THE EIGENIMAGE ADMITTANCE MATRIX A new mathematical model [1] is proposed for solving power flow problem using augmented rectangular system with the system eigen-image admittance matrix, a method to create a mathematical model with a rectangular jacobian matrix consisting of its almost unchanged elements and also to receive a kind of diagonal jacobian matrix for the simplicity of the process of calculation The results of its application prove the advantages of this new proposed model by comparison with the another B11 2B12 2B21 a2 B22 : : : : 2B m1 2B m2 2B1m 2B2m : : : : am B mm P1g P2g : : : Pmg b1 b2 : ; : : bm where Bij denotes the elements of a loss coefficient matrix which can be found from [7] (19) Bij i j) Vi Vj cos cos( i cos N Nb Vi sin i Bm ) ; Im[J iT ] (t m 1)(Vj cos jBm (16) a1 Vj sin jBm ) (15) (t 2m 1)(Vi cos i Bm (18) In case of branch m (linking two buses i, j) which characterized branch’s transfo-ratio different from one unit, the bus voltages depends upon the bus currents so the injected transfo-model currents can be added to the bus current, and we can compute the injected transfo-model currents as follow (14) models for the power flow solution Applying this power flow model to p_power optimization we can develop the following equations C ks J s s j N C ki J J k C ks J s s C kjJ ; where m is the generator bus quantity; N is the load bus quantity; Nb is the branch quantity; Cks, Cki, Ckj are the elements of current distribution factor matrix; Js is the load current at bus s; J is the sum of generator currents; Vi,Vj are the generator voltages; i, j are the angles between current and voltage of generators i, j; cos i, cos j are the power factors of generators i, j The process of power system steady-state optimization implemented by search-method, as the modified zbus model, can be also refered to flow-chart shown on figure Using these algorithms to solve load flow problem we obtain the same result of bus voltages in overall power system The main difference is to compare the influence on the optimization results of two different forms of equations (2) and (18) with the different expressions of loss coefficients (3),(4),(19), involved in determining the technico-economic equivalent states [2] by calculating the R magnitude which determines the proximity state level to the optimum point (20) R m i Pi2 ; where Pi is the p_power deviations from optimum p_power magnitude Pi (i=1,2,…,m) Assume that the existing satisfy to an inequality E1 m (21) B11 Pi2 i a1 B21 B22 Bm1 R2 ; where Cmax (28) HR2 ; where H is a number which must be found to appreciate the sytem fuel cost increment If a maximum eigenvalue max of the estimative matrix is computed, then it will be accepted [5] as the number H in (22) (23) H= m C i B1m B2 m am Bmm ; m Boi ; Pi ( Pi 0); Once the largest eingenvalue 1max of the matrix E1 is found, we can determine the system fuel cost increment (29) C m max i Pi2 ; EXAMPLE OF NUMERICAL RESULTS m max 2BijPLj j max in this case we have (24) Bm then the system fuel cost increment is (22) B12 a2 i 4.1 Example Pi2 ; the system fuel cost increment C is determined as a quadratic function of the generator p_power output deviations Pi with an estimative matrix E Let’s investigate a 220kV power system consisting of generators and of loads The power system’s schema is shown in figure (25) E B11 a1 B21 B12 B22 a B1m B2 m Bm1 Bm Bmm ; am in general, the generator bus quantity n is less than the bus total quantity in the whole power system The loss coefficient matrix B must be a square matrix of (m×m) size as the form of (19) Applying the (3) and (4), we can obtain the modified expression of the system fuel cost increment as follow (26) C m m Bij Pj Pi i 1j m i i Pi2 m a i Pi2 ; i in case of (26), to estimate the equivalent state, we propose a new modified expression of estimative matrix E1 to stand for the matrix E (27) The linedata is given in table The numerical results of initial state parameters calculation are presented in table The modal matrix of the estimative matrix is The eigenvalues are The generator fuel cost functions are given as follow C1=0.007P21g+3.9P1g+155, $/h; (50≤P1g≤275)MW; C2=0.006P22g+4.5P2g+149, $/h; (50≤P2g≤250)MW; C3=0.008P23g+3.2P3g+145, $/h; (50≤P3g≤300)MW; C7=0.0065P27g+3.5P7g+147, $/h; (50≤P7g≤300)MW; Using the augmented regtangular system model (ARSM) with eigen-image matrix to carry out the power system steady-state optimization, we obtain the main numerical results of the problem of economic p_power generation as presented in table And we have 0084; Using modified zbus model (MZbM) to optimize the power system steady-state, analogically, we have the main numerical results of the problem of economic p_power generation as presented in table The loss coefficient sub-matrix is The loss coefficient matrix is the b the system incremental =6.8957; ARSM fuel cost is system incremental =7.15218; fuel cost is Using the modified zbus model, we obtain the estimative matrix E1 as follow In this case the estimative matrix E is The modal matrix of the estimative matrix E1 is 4.2 Example Let’s investigate a standard IEEE 30bus power system consisting of generators The power system’s schema is shown in figure In this case the eigenvalue matrix is And we obtain H1 = 0013; Applying (24) to estimate fuel cost increment, we obtain CARSM=0.13$/h Using (29) to estimate fuel cost increment, we obtain approximative CMZbM=0.2027$/h It means that both optimum results corresponding to their mathematical models are technicoeconomic equivalent Comparing the economic state parameters of a power system loading 825MW, such as incremental fuel costs xxx, fuel costs Cxxx and transmission p_power losses Pxxx , from the results of the two models above, we notice very little differences between their magnitudes The numerical results are follow C = CARSM – CMZbM = 0.052 $/MWh; = ARSM – MZbM = -0.256 $/MWh; d P = PARSM – PMZbM = 0.046 MW; If the maximum system fuel cost increment is accepted in range of 1$/h, then the technicoeconomic equivalent states must be found in a globoil space (precision of economic equivalent state) with radius R=10.91; Comparing the optimum generator p_power outputs of two cases presented above, we can obtain the differences as follow │ │ │ │ P1│=│196.09 –193.57│= │ 2.52│< R; P2│=│187.12 –188.08│= │-1.96│< R; P3│=│216.76 –218.90│= │-2.24│< R; P7│=│239.17– 238.55│= │ 0.62│< R; The numerical results presented in the table and table show the voltage state parameters set together are almost unchanged Investigating the economic equivalent state, we have the numrical results as follow The technico-economic equivalent parameters are shown in table The ARM model estimative parameters are the number H is 0.2461, and the system fuel cost increment is CARSM=10.6468$/h; The MZb model estimative parameters are the number H1 is 0.2394, and the system fuel cost increment is CMZbM=10.3570$/h; Comparing the system fuel cost increment C, we can conclude the both optimum results corresponding to their mathematical models are technico-economic equivalent CONCLUSION The modified zbus power flow model is developed as (5)-(6)-…-(15)-(16)-(17) A new augmented regtangular system power flow model [1] is proposed to power flow optimization with new expression of loss coefficient [7] This paper develops a method estimating the technico-economic equivalent state [2] with an estimative matrix which is determined by a square and symmetric loss coefficient matrix of (n×n) size in which n is the generator quantity of a power system, including the slack bus generator [7] A new loss coefficient expression [3] and a new modified expression (27) of estimative number H are proposed Using the H and R magnitudes we can analyse the influence of power system parameters on the technicoeconomic equivalent state in electrical power system operation with an estimative matrix E obtained from the results of a power system state optimization solving REFERENCES Luu Huu Vinh Quang Electrical Power System steady-state simulation using Augmented Rectangular System Model with the Eigen-Image Admittance Matrix Authorship testimonial No1175/2004 /QTG Copyright Office of VN Luu Huu Vinh Quang A method for estimating the technico-economic equivalent states in power system operation th Proceedings of the 10 conference on science and technology - University of Technology VNU HCMC - 2007 International Advanced Symposium on Electrical Engineering Luu Huu Vinh Quang Economic p_power dispaching and load q_power compensation in power system Authorship testimonial No63-15-3-2001- Copyright Office of VN Glenn W Stagg, Ahmed H.El-Abiad Computer method in power system analysis McGraw-Hill 1983 V.A Venikov Electrical Power System Computing Programming and Optimization Moscow High School Edition 1973 John J.Grainger & William D.Stevenson,JR Power system analysis McGraw-Hill 1994 Luu Huu Vinh Quang A new P_Power losscoefficient expression and economic active power generation of thermal plans th Proceedings of the conference on science and technology p.1-6 Session Electrical Engineering and Power systems University of Technology VNU HCMC Hadi Saadat Power system analysis McGraw-Hill 1999